On the Accuracy of the Numerical Detection of Complex Obstacles from Far Field Data Using the Probe Method

Abstract

We deal with the acoustic inverse scattering problem for detecting an obstacle with mixed boundary conditions from the far field map. We show how the geometrical properties and the material parameter distributed on the surface are involved in the obstacle reconstruction numerically. The main advance of this research is the numerical analysis and implementation of our recent theoretical work [J.J. Liu and M. Sini, How to make the reconstruction of obstacles more (or less) accurate from exterior measurements, RICAM preprint 2008-04, Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria, 2008] regarding the higher order asymptotic expansion of the probe indicator function and the introduction of complex-valued surface impedance. An efficient numerical implementation scheme is proposed based on the properties of minimum norm solutions. Precisely, using the relation between the surface impedance and the obstacle curvature contained in the higher order expansion of the indicators, we reveal how the obstacle curvature and the surface impedance on the coated part influence the blowing-up behavior. Using this theoretical result, we can specify the complex surface impedance in terms of the obstacle curvature to make the reconstruction more (or less) accurate. By establishing properties of the minimum norm solution for approximating the singular sources, efficient realizations for approximating the multipoles by the Herglotz wave function and therefore the implementations of the probe methods are developed with an error estimate. We finally show extensive numerical tests explaining how and to what extent the coupling relation between the curvature and the surface impedance changes the accuracy of the reconstruction of the obstacles.

On the reconstruction of Dirichlet-to-Neumann map in inverse scattering problems with stability estimates

“…and the curve Γ is chosen to guarantee the solvability of (14) and (16) given in the sequel. Then it follows for x, y ∈ ∂Ω that…”

“…In the original version of this scheme, the indicator is constructed by firstly computing the D-to-N map (5) using the far-field data for suitably chosen Ω and boundary value f . Here, we would like to point out that the indicator in the probe method can also be constructed directly from the far-field pattern in one step [15], which is firstly noticed in [17] and then is developed in [16]. For other reconstruction schemes for the obstacle with impedance boundary, we refer to [3] and the references therein.…”

“…These numerical implementations did not involve the process of reconstructing Dto-N map from the far-field data, which is the first step in the initial version of probe method using the physical measurement data. Although the probe method can use an indicator constructed from the far-field data directly as done in [15][16][17], the original version based on the D-to-N map is still of great importance for the thorough understanding on the probe method, i.e., the probe method essentially extracts the information about the Green function of the boundary value problem from the far field data to reconstruct the obstacle boundary. Moreover, to the authors' knowledge, the numerical implementations of D-to-N map from the far-field pattern have not yet been realized efficiently up to now.…”

“…For given bounded domains Ω and Γ, assume that both(14) and(16) are solvable. Let B(R, x 0 ) be a suitably chosen disk satisfying Γ ⊂⊂ B(R, x 0 ) ⊂⊂ Ω with radius R and center x 0 .…”

Recovering the Dirichlet-to-Neumann map in inverse scattering problems using integral equation methods

“…More explicitly, in the original version of the probe method, the indicator function is constructed by firstly computing the D-to-N map (1.5) for unknown D from the far-field data {w ∞ (θ, d) : θ, d ∈ S}, with suitable choices of and f related to some detection point z outside D. We then get the profile of the scatterer by capturing the blow-up behavior of the indicator function as z → ∂ D. If the scatterer is an obstacle with the boundary of impedance type, then we can also recover the surface impedance from the Dto-N map by moment method as suggested in [1]. Hence, the reconstruction of the D-to-N map from the far-field measurements is very interesting and important in inverse scattering problems, although the indicator function can be constructed directly from the far-field data as done in [12][13][14]. Using Green's representation theorem and properties of layer potentials, this problem can be reduced to solving an integral equation of the second kind, where the kernels are singular and involve the derivatives of the scattered waves for point sources.…”